Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{t^2 - 36}{t - 6}$
Explanation: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = t$ $ b = \sqrt{36} = -6$ So we can rewrite the expression as: $z = \dfrac{({t} {-6})({t} + {6})} {t - 6} $ We can divide the numerator and denominator by $(t - 6)$ on condition that $t \neq 6$ Therefore $z = t + 6; t \neq 6$